Download Orbital Physics Part 1: Gravitational Potential Energy Our start will be

Orbital Physics
Part 1: Gravitational Potential Energy
Our start will be potential energy but what is that?
The word potential in potential energy indicates that in some fashion this is an
unrealized energy but also that this unrealized energy has some possibility of being
converted into a more active or familiar form.
One example of a potential energy is the chemical energy in gasoline. This is the
energy that is contained in the bonds between atoms in the molecules that form
gasoline. Every gallon of gasoline contains an amount of this potential energy that could
be released if the gasoline is burned, for example if someone throws a lit match into a
gallon container of gasoline. A gallon of water clearly does not have the same potential.
We could refer loosely to these differences as a difference in the chemical potentials of
the gasoline and water even though without any lit matches flying around none of this
potential will be realized.
A potential energy that we are more familiar with is the gravitational potential energy,
although you may not at first realize it.
Suppose we have a book and hold it suspended a meter above the floor. The book has
a certain gravitational potential energy relative to the floor. That energy is realized if we
let go of the book and let it fall. By the time the book reaches the floor it will have a
certain amount of kinetic energy. This kinetic energy, neglecting air resistance, will be
equivalent to the gravitational potential energy that the book had before we dropped it.
During the fall, the gravitational potential energy was transformed into a more active
form of energy, kinetic energy.
This is why we have the concept of potential energy. In energy terms, having one book
on the floor and an identical one held 1 meter above the floor is not the same even
though we see no energy differences unless we release the books. We describe these
differences in terms of potential energy.
Holding a book 2 meters above the floor can have a different result if we let go of it than
holding a book of equal mass 1 meter above the floor and release it there. The two
books will not hit the floor with the same amount of kinetic energy so we recognize this
possibility, before their actual release, by saying that they have different gravitational
potential energies.
The formula for gravitational potential energy is mgh or an objects weight multiplied by
its height. Increasing the height of the object increases the gravitational potential energy
that the object has, something you can probably directly appreciate if the object is you
and you are climbing a tree. Your gravitational potential energy increases as you move
higher in the tree, reflecting the increased kinetic energy and chanced of injury you will
have if you fall and hit the ground.
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To make a determination of gravitational potential energy we have to have a reference
height from which to measure our vertical distances. In the case of climbing a tree
the obvious place to set a reference level where gravitational potential energy is 0 is the
ground since that is the point at which, if we fall, all our potential energy is converted to
kinetic energy (and potential energy is, for practical purposes, equal to zero).
Whenever measuring gravitational potential energy we need such a reference point but
there is no requirement as to where we have to put it. We can choose whatever height
is convenient. The reason we can do this is that the important thing about a fall is the
distance fallen, not what the heights are at the start and finish. For example, if we fall in
the mountains from a height of 4,000 feet to an elevation of 3,900 feet the result is the
same as if we fell from an elevation 3,000 feet to an elevation of 2,900 feet as far as the
kinetic energy we have at the end of the fall.
Now suppose we have a book that weighs 10 Newtons (W = mg) and we want to use it
to map out all the points above the ground where the gravitational potential energy is 10
Joules. We set the reference level of 0 gravitational potential energy at the level ground.
Gravitational potential energy is the object’s weight multiplied by its height above the
reference level, making the 10 Joule (1 Joule = 1 Newton-m) gravitational potential
energy level everywhere 1 meter above the ground.
We call this level where the gravitational potential energy is the same an equipotential
surface.
The 20 Joule equipotential surface is then 2 meters above the ground and so on. If we
keep going up we keep coming to and crossing equipotential surfaces of increasing
gravitational potential energy.
We already know that these equipotential surfaces are just as level as the reference
height of the level ground but if we back away from the Earth and take a larger view we
see that this is not entirely accurate. The surface of the Earth is actually curved and the
equipotential surfaces are curved as well:
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If we view the Earth as a whole, we find that the equipotential surfaces are concentric
with the surface and form circles:
If we move across these surfaces away from the Earth our gravitational potential
energy increases, if we move across these surfaces closer to the Earth our gravitational
potential energy decreases.
We can make the same argument relative to the gravitational field of the Sun, in which
case the handiest reference level for gravitational potential energy is the surface of the
Sun. The equipotential surfaces are then concentric with the surface of the Sun:
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….with the lower energy surfaces closer to the Sun and the higher energy surfaces
farther from the Sun (in essence as we move away from the Sun we are climbing in its
gravity field, increasing our gravitational potential energy relative to the Sun and if we
move toward the Sun we are falling toward the Sun, falling in its gravity field, decreasing
our gravitational potential energy relative to the Sun.
Now let’s superimpose an elliptical planetary orbit on this energy level diagram:
Now we are going to find out what gravitational potential energy has to do with orbital
dynamics.
Gravity is a special type of force called a conservative force. When there are no
dissipative forces present, friction being the principal dissipative force, a condition holds
with the conservative force called the conservation of mechanical energy.
In space, friction is essentially zero so with gravity being the operational force we have
the conservation of mechanical energy. This means that the sum of the gravitational
potential energy and kinetic energy of a planet are, everywhere in the orbit, equal to a
constant. This, in turn, means that any decrease in gravitational potential energy must
be balanced by an equal increase in the kinetic energy and vice versa.
Remember that gravitational potential energy depends on the height above the surface
of the Sun, which is the distance between the planet and the Sun. Gravitational potential
energy is highest when the planet is farthest from the Sun (aphelion) and decreases as
a planet moves away from aphelion, reaching a minimum at the closest point in the
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planet’s orbit to the Sun (perihelion) and then increases again as the planet moves
away from perihelion toward aphelion again.
Due to the conservation of mechanical energy, kinetic energy shows the opposite
pattern: when gravitational potential energy is highest, at aphelion, kinetic energy is at a
minimum. As the planet moves away from aphelion, its gravitational potential energy
drops so its kinetic energy must rise. Gravitational potential energy reaches a minimum
at perihelion so this is where the kinetic energy of the planet is greatest. As the planet
moves away from perihelion back out toward aphelion, its gravitational potential energy
rises and its kinetic energy must fall.
The formula for kinetic energy is mv2/2. At the speeds at which planets orbit in the solar
system, mass is essentially a constant so the only way kinetic energy can vary is for the
velocity to change. If the kinetic energy of a planet goes up it move faster and if its
kinetic energy goes down it moves more slowly.
The interaction of gravitational potential energy and kinetic energy of an orbiting planet
in an elliptical orbit means that the closer to the Sun the planet is the faster its
orbital speed must be and the farther away it is the slower its orbital speed must
be. Because mechanical energy is conserved in planetary orbits the orbital speed
of a planet depends on its distance from the Sun. The closer a planet gets to the
Sun the faster it moves in its orbit and the farther away a planet gets from the Sun
the slower it moves in its orbit.
Conservation of mechanical energy explains why planets in elliptical orbits do
not always orbit at the same speed.
A planet in a circular orbit will always be the same distance from the Sun so its orbital
speed will not change. However, all the planets in our solar system orbit the Sun in
elliptical orbits with the Sun at one focus of the ellipse. This means that the distance of
any planet in our solar system from the Sun does not remain constant and neither does
its orbital speed.
We can apply the same arguments about energy to just about anything orbiting just
about anything else, including satellites orbiting the Earth, so this is a very useful type of
analysis. Notice also that we did not have to deal with vectors, even though motion was
involved. This scalar property is a general feature of energy and often makes solving a
problem using energies simpler than another mode of solution that requires vectors.
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Condition required for Conservation of Mechanical Energy:
There must be no dissipative forces (such as friction). Dissipative forces dissipate
energy in ways that make recovery impossible.
Equations of Gravitational Conservation of Mechanical Energy:
PEG = mgh = Wh
PEG = gravitational potential energy
m = mass
g = gravitational acceleration (gravity field)
h = height above a reference level where h = 0
W = weight = mg
KE = mv2/2
KE = kinetic energy (energy of movement)
m = mass
v = velocity
PEG + KE = mgh + mv2/2 = constant
∆PEG = -∆KE
The above two equations are the mathematical statement of the conservation of
mechanical energy.
∆PEG = change in gravitational potential energy
∆KE = change in kinetic energy
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